Metric, density coefficients, coordinates, tests

With $H={\mathrm{tr}}_{\eta }h$,

so that $\sqrt{-q}{q}^{\mu \nu }={\eta }^{\mu \nu }+\kappa a^{\mu \nu }+{\kappa }^2{A}^{\mu \nu }+O({\kappa }^3)$ and $\sqrt{-q}=1+\kappa b+{\kappa }^{2}B+O({\kappa }^3)$.

The ramp coordinates ${\mathsf{X}}_1,{\mathsf{X}}_2$ have zero Cauchy data at $\Sigma =\{t=0\}$ and solve

For sources vanishing on $t\le 0$ this agrees with retarded propagation, ${\mathsf{X}}_1=-{\partial }_{\mu }{{\mathsf{E}}_0}^{\mathrm{R}}({\sigma }_{\epsilon }{a}^{\mu \rho })$. The bare tests $c_1={\partial }_{\mu }(f{{\mathsf{X}}_1}^{\mu })$, $c_2$, the Euler coefficients

and the completed tests

are X-D's, at the fixed $\epsilon$. The completed scalar-only observable is $\hat{\Phi }(f)=\hat{\varphi }(f)+\kappa \hat{\varphi }(d_1)+{\kappa }^2\hat{\varphi }(d_2)$. We write